My question is regarding the definition $3.2$ of the book "groups acting on graphs" by Dicks and Dunwoody.
The notation $G\otimes_H U$ was defined in the definition $3.1$. Let me review it here.
Let $H$ is a subgroup of $G$ and $U$ be $H$-set. Then $G\otimes_H U$ denotes the set $G\times U$ obtained by identifying $(gh,u)=(g,hu)$.
In the definition $3.2.$:
They assume that $T$ is $G$-tree and $U$ a $G$-transversal in V(T). In addition they suppose that for each $u\in U$ there is a $G_u$-tree($G_u$ is the stabilizer of the vertex $u$).
Then they consider $G$-forest $\bigcup_{u\in U} G\otimes_{G_u}T_u$.
I cannot see how they define a graphical structure on $G\otimes_{G_u}T_u$. More precisely, how can we define an edge between $g_1\otimes u_1$ and $g_2\otimes u_2$, where $g_i\in G$ and $u_i\in T_u$?
Let me use the notation $[g,u] \in G \otimes_H U$ for the equivalence class of $(g,u)$ (what you denoted $g \otimes u$).
The definition of vertices in $G \otimes_{G_u} T_u$ is simply that for each vertex $P \in V(T_u)$ and each $g \in G$ we have a vertex $[g,P] \in V(G \otimes_{G_u} T_u)$.
And the definition of edges in $G \otimes_{G_u} T_u$ is that for each edge $E$ in $T_u$ having endpoints $P,Q \in V(T_u)$, and for each $g \in G$, there is an edge $[g,E]$ in $G \otimes_{G_u} T_u$ having endpoints $[g,P]$, $[g,Q]$.